Zeno's Paradoxes and the Infinite Sum

Achilles races a tortoise and gives it a head start. By the time Achilles reaches where the tortoise was, it has moved ahead. By the time he reaches that new point, it has moved again — forever. So Achilles can never overtake the tortoise. Zeno of Elea posed this paradox around 450 BCE. The resolution — which took 2,000 years to make rigorous — is that an infinite number of steps can take a finite amount of time: 1/2 + 1/4 + 1/8 + … = 1. The mathematics of convergent infinite series resolves what seemed like a logical impossibility.